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MATH 246 Probability and Random Processes
Final Examination
Fall 2004 Course Instructor: Prof. Y. K. Kwok

Time allowed: 100 minutes
1. A miner is trapped in a mine containing 3 doors. The .rst door leads to a tunnel that will take him to safety after 3 hours of travel. The second door leads to a tunnel that will return him to the mine after 5 hours of travel. The third door leads to a tunnel that will return him to the mine after 7 hours. If we assume that the miner is at all times equally likely to choose any one of the doors, what is the expected length of time until he reaches safety? [4]
Hint: Let X be the amount of time (in hours) until the miner reaches safety, and let Y denote the door
he initially chooses. By the rule of conditional expectation

E[X]= E[X|Y = 1]P [Y = 1]+ E[X|Y = 2]P [Y = 2]
+ E[X|Y = 3]P [Y = 3].
2. Consider the pair of random variables X and Y whose joint density function is given by
. 1 x2 + y2 1
fXY (x, y)= .
0 otherwise
Show that X and Y are uncorrelated. Are they independent? [6]
3. The time between consecutive earthquakes in San Francisco and the time between consecutive earth-11

quakes in Los Angeles are independent and exponentially distributed with means and , respec-
1 2
tively. What is the probability that the next earthquake occurs in Los Angeles? [7]
4. Let X and Y be a pair of independent random variables, where X is uniformly distributed over (.1, 1) and Y is uniformly distributed over (0, 2). Find the probability density of Z = X/Y . [8]
Hint: Explain why
fZ (z)= |y|fXY (yz, y) dy.
The region {(y, z): .1 <yz < 1 and 0 <y< 2} can be divided into 3 regions, according to (i) 111 1
z< . , (ii) . z and (iii) z> .
222 2
5. Let N(t), t 0, be a Poisson process with parameter > 0. (a) Show that the autocovariance CN (t1, t2) of N(t) is given by
CN (t1, t2) = min(t1, t2).
In your derivation steps, explain clearly how you use the stationary increments and independent increments properties of a Poisson process. (b) Suppose a Poisson event is known to have occurred over the time period [0, 1], show that the probability of the event occurring before time t, 0 < t < 1, is equal to t. Hint: Consider [5] [5]
P [N(t) = 1|N(1) = 1], where 0 < t < 1.
6. Let X(t) = A cos t + B sin t, where A and B are independent and identically distributed Gaussian random variables with zero mean and variance 2 . Find the mean and autocovariance of X(t). [6]
7. A machine consists of two parts that fail and are repaired independently. A working part fails during any given day with probability . A part that is not working is repaired by the next day with probability . Let Xn be the number of working parts in day n. The sample space of Xn is {0, 1, 2}. We write n,j = P [Xn = j], j = 0, 1, 2.
(a) Find the one-step transition probability matrix P , expressed in terms of and . state pmf vector is If the initial